Consider two congruent circles $\odot A$ and $\odot B,$ and a point on each one.

Because $\odot A\cong \odot B,$ the distance from the center of the circle to a point on the circle is the same for each circle. Therefore, $\overline{AC}\cong \overline{BD}$ which implies that both circles have the same radius.

Conversely, consider two circles with the same radius.

Next, translate $\odot B$ so that point $B$ maps to point $A.$ The image of $\odot B$ is $\odot B^{\prime }$ which is a circle centered at $A.$ Since the circles have the same radius, this translation maps $\odot B$ onto $\odot A.$

Because a rigid motion maps one circle onto the other, it is concluded that both circles are congruent.